Geometric and combinatorial aspects of commutative algebra

geometric and combinatorial aspects of commutative algebra

geometric and combinatorial aspects of commutative algebra

edited by Jürgen Herzog Universität Essen Essen, Germany

Gaetana Restuccia Università di Messina Messina, Italy

Marcel Dekker, Inc.

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PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universität Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

N. Jacobson, Exceptional Lie Algebras L.-Å. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis I. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et al., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et al., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et al., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thomier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L. Cesari et al., Nonlinear Functional Analysis and Differential Equations J. J. Schäffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et al., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et al., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Time Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L. Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces

61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122.

O. A. Nielson, Direct Integral Theory J. E. Smith et al., Ordered Groups J. Cronin, Mathematics of Cell Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L. Herdman et al., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Geel, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A. Fiacco, Mathematical Programming with Data Perturbations I P. Schultz et al., Algebraic Structures and Applications L Bican et al., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et al., Volterra and Functional Differential Equations N. L. Johnson et al., Finite Geometries G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urruty et al., Optimization A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istra'tescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathematical Logic and Formal Systems C. E. Aull, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L. Salce, Modules Over Valuation Domains P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations C. A. Baker and L. M. Batten, Finite Geometrics J. W. Brewer et al., Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science B.-L. Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M. C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations P. Clément et al., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et al., Differential Equations E. O. Roxin, Modern Optimal Control J. C. Díaz, Mathematics for Large Scale Computing Ú P. S. Milojevic, Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations

123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.

R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics W. D. Wallis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Parameter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Arino et al., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et al., General Topology and Applications P Clément et al., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry I. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang–Mills Connections L. Fuchs and R. Göbel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clément and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et al., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolésio, Boundary Control and Variation M. Kr'íz''ek et al., Finite Element Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et al., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Aglianò, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

184. J. E. Andersen et al., Geometry and Physics

185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233.

P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et al., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L. Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm–Liouville Problems W. H. Schikhof et al., p-adic Functional Analysis S. Sertöz, Algebraic Geometry G. Caristi and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Kr'íz''ek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et al., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology Z. Chen et al., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yéldérém and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et al., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et al., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators

Additional Volumes in Preparation

Preface

This volume contains research papers based on lectures delivered at the International Conference "Commutative Algebra and Algebraic Geometry," held in Messina, Italy. Altogether 116 mathematicians from Europe, the United States, Japan, India, Korea, Brazil, Senegal, and from all parts of Italy attended the meeting. One traditional research subject at the University of Messina is commutative algebra. The motivation to organize this meeting was to celebrate this tradition and to further promote this area of mathematics in Sicily. In recent years combinatorial and computational methods found successful applications in commutative algebra and algebraic geometry. One aim of this book is to focus on this development, but also to present new trends in singularity and tight closure theory, the theory of projective schemes and the geometry of curves in Pn. The present volume is directed to university teachers as well as to graduate students and is intended to stimulate further research in these topics. We would like to thank many public and private institutions for their support and enthusiasm, which made it possible to organize this congress, among them the MURST (National project Algebra Commutativa Geometria Algebrica e Aspetti Computazionali), the University of Messina, the Presidenza della Regione Siciliana, the CNR, the Fondation Bonino Pulejo of Messina, and the Faculty of Ingineering of the University of Reggio Calabria. We also would like to thank the Faculty of Sciences of the University of Messina for providing the required facilities during the conference, and the Centre di Spiritualita S. Tommaso di Messina for granting hospitality to many participants of the conference. We are grateful to the Scientific Committee consisting of Prof. Takayuki Hibi (Osaka University), Prof. Lawrence Ein (Illinois University), Prof. Hamet Seydi (Universite de Dakar), Prof. Rosario Strano (University of Catania), Prof. Paolo Valabrega (Politecnico di Torino), and Prof. Wolmer Vasconcelos (Rutgers University) for its cooperation, and we wish to thank all contributors and participants for their active collaboration, as well as all the referees for their help. Our thanks go also to all local organizers: Vittoria Bonanzinga, Marilena Crupi, Maurizio Imbesi, Rosanna Utano and the students of the doctorat Sabina Buono, Anna Maria Giordano, Maria Fortuna Paratore, and Giancarlo Rinaldo for their help during the meeting. In particular, we would like to thank iii

Preface

Giancarlo Rinaldo for his help in the preparation of the meeting. Special thanks go to Rosanna Utano, who spent a lot of energy and passion in the preparation of this book.

Jiirgen Herzog

Gaetana Restuccia

Contents

Preface Contributors Participants 1.

On Free Complete Intersections Margherita Barile, Marcel Morales, and Apostolos Thorna

2.

On the Number of Ideals of Finite Colength Valentina Barucci and RalfFroberg

3.

Realizable Sequences Linked to the Hilbert Function of a 0-Dimensional Protective Scheme Giannina Beccari and Carlo Massaza

4.

5.

iii ix xiii 1

11

21

Lexsegment Ideals in the Exterior Algebra Vittoria Bonanzinga

43

On the Equations Defining Toric Projective Varieties

57

Emilio Briales Morales, Antonio Campillo Lopez, and Pilar Pison Casares 6.

KRS and Determinantal Ideals Winfried Bruns and Aldo Conca

67

1.

The Koszul Complex in Projective Dimension One Winfried Bruns and Udo Vetter

89

8.

Grobner Bases as Characteristic Sets Giuseppa Carra'Ferro

99

9.

Threefolds with Degenerate Secant Variety: On a Theorem of G. Scorza L. Chiantini and C. Ciliberto

10. The Unirationality of All Conic Bundles Implies the Unirationality of the Quartic Threefold

Alberto Conte (with an Appendix by M. Marchisio)

111

125

Contents

11.

Gaussian Ideals and the Dedekind-Mertens Lemma Alberto Corso and Sarah Glaz

131

12. Depth Formulas for Certain Graded Modules Associated to a Filtration: A Survey T. Cortadellas and S. Zarzuela

145

13. Extremal Betti Numbers of Lexsegment Ideals

159

Marilena Crupi and Rosanna Utano 14. Local Monomialization Steven Dale Cutkosky

165

15.

175

Isomorphism of Complexes and Lifts Steven Dale Cutkosky and Hema Srinivasan

16. Weil Divisors on Rational Normal Scrolls Rita Ferraro

183

17. Families of Wronskian Correspondences

199

Letterio Gatto 18. A Note on Hodge's Postulation Formula for Schubert Varieties Sudhir R. Ghorpade

211

19.

221

Unimodular Rows and Subintegral Extensions

Joseph Gubeladze 20. On Commutative FGS-Rings with Ascending Condition on Annihilators

227

Cheikh Thiecoumba Gueye and S. Mamadou Sanghare 21.

Cohen-Macaulay F-Injective Homomorphisms Mitsuyasu Hashimoto

231

22. Valuations of Ideals, Evolutions and the Vanishing of Cohomology Groups Reinhold Hiibl and Anton Rechenauer

245

23.

259

On Integral Schemes of Vector Fields Ernst Kunz

24. Roberts Rings and Dutta Multiplicities

273

Kazuhiko Kurano 25. Hilbert Functions of Squarefree Veronese Subrings Hiroyuki Nishida, Hidefumi Ohsugi, and Takayuki Hibi 26.

On the Conductor of a Surface at a Point Whose Projectivized Tangent Cone Is a Generic Union of Lines Ferruccio Orecchia

289

301

Contents

vii

27. Lifting Problem for Codimension Two Subvarieties in Pn+2: Border Cases Margherita Roggero

309

28. Rank Two Bundles and Reflexive Sheaves on P3 and Corresponding Curves: An Overview Margherita Roggero, Paolo Valabrega, and Mario Valenzano

327

29. On the Structure of Ext Groups of Strongly Stable Ideals Enrico Sbarra

345

30. An Introduction to Tight Closure

353

Karen E. Smith 31.

Eisenbud-Goto Inequality for Stanley-Reisner Rings Naoki Terai

379

Contributors

Margherita Barile Universita degli Studi di Bari, Bari, Italy Valentina Barucci Universita "La Sapienza," Rome, Italy Giannina Beccari Politecnico di Torino, Torino, Italy Vittoria Bonanzinga University of Reggio Calabria, Reggio Calabria, Italy Winfried Bruns University of Osnabriick, Osnabriick, Germany Antonio Campillo Lopez

University of Valladolid, Valladolid, Spain

Giuseppa Carra' Ferro University of Catania, Catania, Italy Pilar Pison Casares University of Seville, Seville, Spain

L. Chiantini University of Siena, Siena, Italy C. Ciliberto Universita di Roma Tor Vergata, Rome, Italy Aldo Conca University of Sassari, Sassari, Italy Alberto Conte Universita degli Studi di Torino, Torino, Italy Alberto Corso University of Kentucky, Lexington, Kentucky T. Cortadellas University of Barcelona, Barcelona, Spain Marilena Crupi University of Messina, Messina, Italy Steven Dale Cutkosky University of Missouri-Columbia, Columbia, Missouri Rita Ferraro Universita di Roma "Tor Vergata," Rome, Italy

Ralf Froberg Stockholms Universitet, Stockholm, Sweden Letterio Gatto Politecnico di Torino, Torino, Italy ix

x

Contributors

Sudhir R. Ghorpade Sarah Glaz

Indian Institute of Technology, Bombay, India

University of Connecticut, Storrs, Connecticut

Joseph Gubeladze A. Razmadze Mathematical Institute, Tbilisi, Georgia Cheikh Thiecoumba Gueye

Universite Cheikh Anta Diop, Dakar, Senegal

Mitsuyasu Hashimoto Nagoya University, Nagoya, Japan Takayuki Hibi

Osaka University, Osaka, Japan

Reinhold Hiibl

University of Regensburg, Regensburg, Germany

Ernst Kunz

University of Regensburg, Regensburg, Germany

Kazuhiko Kurano Tokyo Metropolitan University, Tokyo, Japan Carla Massaza

Politecnico di Torino, Torino, Italy

Eniilio Briales Morales

Marcel Morales

University of Sevilla, Seville, Spain

University of Grenoble I, Institut Fourier, Lyon, France

Hiroyuki Nishida

Osaka University, Osaka, Japan

Hidefumi Ohsugi Osaka University, Osaka, Japan Ferruccio Orecchia

Complesso Universitario di Monte S. Angelo, Naples,

Italy Anton Rechenauer University of Regensburg, Regensburg, Germany Margherita Roggero University of Torino, Torino, Italy

Mamadou Sanghare Enrico Sbarra

Universite Cheikh Anta Diop, Dakar, Senegal

Universitat GH-essen, Essen, Germany

Karen E. Smith University of Michigan, Ann Arbor, Michigan Hema Srinivasan University of Missouri—Columbia, Columbia, Missouri

Naoki Terai Faculty of Culture and Education, Saga University, Saga, Japan Apostolos Thoma University of loannina, loannina, Greece Paolo Valabrega Politecnico di Torino, Torino, Italy Mario Valenzano Politecnico di Torino, Torino, Italy

Contributors

Udo Vetter Universitat Oldenburg, Oldenburg, Germany S. Zarzuela University of Barcelona, Barcelona, Spain

xi

Participants

• ARAMOVA Annetta Universitat Essen, Germany, [email protected]

• ARENA Roberto Via Raimondi, 17 - Messina, Italy, [email protected]

• ARTALE Maria Universita di Roma, Tor Vergata, Italy, [email protected]

• BARILE Margherita Universita di Bari, Italy, [email protected] • BARRY Mamadou Dept. Maths et Info. FST, Universite de Dakar, Senegal, [email protected]

• BARUCCI Valentina Universita di Roma "La Sapienza", Italy, [email protected]

• BASS Hyman Columbia University, New York - USA, [email protected] • BECCARI Giannina Politecnico di Torino, Italy, [email protected]

• BLICKLE Manuel University of Michigan, USA, [email protected]

• SLUM Stefan Universitat Essen, Germany, [email protected] • BONANZINGA Vittoria Universita di Reggio Calabria, Italy, [email protected], [email protected]

• BRUNS Winfried University of Osnabruck, Germany, [email protected] • BUCHSBAUM David A. Brandeis University, Waltham, MA - USA, [email protected] • BUONO Maria Sabina Universita di Messina, Italy, [email protected]

. CALABRI Alberto Universita di Roma "La Sapienza", Italy, [email protected]

• CALAPSO Maria Teresa

Universita di Messina, Italy, [email protected]

xiii

xiv

Participants

• CARINI Luisa Universita di Palermo, Italy, [email protected] • CARISTI Giuseppe Via Pozzo Giudeo, Torre Faro - Messina, Italy, [email protected]

• CARRA FERRO Giuseppa

Universita di Catania, Italy, [email protected] . CAVALIERE Maria Pia Universita di Geneva, Italy, [email protected]

• CHIANTINI Luca Universita di Siena, Italy, [email protected] • CHO Young Hyun Dept. of Maths, National University, Seoul, Korea, [email protected] . CHRISTENSEN Lars Winther

University of Copenhagen, Denmark, [email protected]

• CILIBERTO Giro Universita di Roma , Tor Vergata, Italy, [email protected]

• CIOLLI Fabio Olevano Romano (Roma), Italy, [email protected]

« CONCA Aldo Universita di Genova, Italy, [email protected] • CONTE Alberto Universita di Torino, Italy, [email protected]

• CONTESSA Maria Universita di Palermo, Italy

• CREA Rosa Giovanna Via Quarnaro, Trav. I Ramirez, 27 - Gallico (RC), Italy

• CRUCITTI Angelica Via Sbarre C.li, Trav. II Ferr., 12 - Reggio Calabria, Italy • CRUPI Marilena Universita di Messina, Italy, [email protected] • CUFARI Adelaide Vill. Annunziata Conca D'Oro, Messina, Italy, [email protected]

• CUMINO Caterina Politecnico di Torino, Italy, [email protected]

Participants • CUTKOSKY Dale S. University of Missouri, USA, [email protected]

• DE FILIPPIS Vincenzo Via Conte D'Apice, 7 - Vibo Valentia, Italy, [email protected]

• DE NEGRI Emanuela Universita di Genova, Italy, [email protected] • DIANKHA Oumar Dept. Maths et Info. FST, Universit de Dakar, Senegal, [email protected]

• DONATO Santi Universita di Messina, Italy • EIN Lawrence Illinois University, [email protected]

• ELIAS Juan Universitat de Barcelona, Spain, [email protected]

• FERRARA Massimiliano Universita di Messina, Italy, [email protected] • FERRARO Maria Rita Universita di Roma, Tor Vergata, Italy, [email protected]

• FIORENTINI Mario Universita di Ferrara, Italy

• FLACCOMIO Simona Via S. Sebastiano, 15 - Messina, Italy, [email protected]

• FRANKILD Anders Matematisk Afdeling, Copenhagen, Denmark, [email protected]

• FROBERG Ralf University of Stockholm, Sweden, [email protected]

• GATTO Letterio Universidade Federal de Pernambuco, Brazil, [email protected], [email protected]

• GERAMITA Anthony V. Universita di Genova, Italy, [email protected] • GERMANA Clara Via S. Caterina dei Bottegai, 8 - Messina, Italy, [email protected]

• GHORPADE Sudhir Indian Institute of Technology - Bombay, India, ghoparde@[email protected]

xv

xvi

Participants

• GIORDANO Anna Maria Universita, di Messina, Italy, [email protected]

• GIUFFRIDA Salvatore Universita di Catania, Italy

• GLAZ Sarah University of Connecticut, USA, [email protected]

• GUARDO Elena Universita, di Catania, Italy

• GUBELADZE Joseph Georgian Academy of Sciences - Tbilisi, Georgia, [email protected]

• GUEYE Cheikh Thiecoumba Dept. Maths et Info. FST, Universit de Dakar, Senegal, [email protected]

• HASHIMOTO Mitsuyasu Nagoya University, Japan, [email protected] . HREINSDOTTIR Freyja Stockholm University, Sweden, [email protected]

. HUBL Reinhold Umversitat Regensburg, Germany, [email protected]

• IARROBINO Anthony [email protected]

• IMBESI Maurizio Universita di Messina, Italy, [email protected] • IONESCU Cristodor Romanian Academy of Sciences, Bucarest, Romania, [email protected] • IORFIDA Vincenzo Contrada Maurizio 10, Lamezia Terme, Italy, [email protected]

• KUNZ Ernst Universitat Regensburg, Germany, [email protected]

• KURANO Kazuhiko Tokyo Metropolitan University, Japan, [email protected]

• LA BARBIERA Monica Via Peschiera, 23 - Messina, Italy

• LI MARZI Enzo Universita di Messina, Italy, [email protected]

Participants

xvii

• MAGGIONI Renato Universita di Catania, Italy, [email protected] • MARCHISIO Marina Universita di Torino, Italy, [email protected]

• MARINARI Maria Grazia Universita di Geneva, Italy, [email protected] • MOLICA BISCI Giovanni Via Passoforno - Piraino (ME), Italy, [email protected]

• NAKAMURA Yukio Meiji University, Higasimita, Japan, [email protected]

• NASTOLD Hans-Joachim Mathematisches Institut, Miinster, Germany, [email protected]

• NISHIMURA Jun-ichi Hokkaido University of Education, Sapporo, Japan, [email protected]

• OHSUGI Hidefumi Osaka University, Japan, [email protected] • ORRCCHIA Ferruccio Universita di Napoli, Italy, [email protected] • PAENG Seong-Hun(non e chiaro se e venuto) Korea Institute for Advanced Study, Seoul, Korea, [email protected] • PANTI Giovanni Universita di Udine, Italy, [email protected] • PARATORE Mariafortuna Universita di Messina, Italy, [email protected]

• PARESCHI Giuseppe Universita di Roma , Tor Vergata, Italy, [email protected] • PAXIA Giuseppe Universita di Catania, Italy • PEDRINI Claudio Universita di Geneva, Italy, [email protected] • PLANAS-VILANOVA Francesc Universitat Politecnicade Catalunya, Barcelona, Spain, [email protected] • POLO Antonio Universita di Bologna, Italy, [email protected]

xviii

Participants

• RAGUSA Alfio Universita di Catania, Italy, [email protected]

• RAVIDA DONATO Maria Via S. Michele, 18 - Castroreale (ME), Italy • RESTUCCIA Gaetana Universita di Messina, Italy, [email protected] • RINALDO Giancarlo Universita di Messina, Italy, [email protected]

• ROBBIANO Lorenzo Universita di Geneva, Italy, [email protected]

• ROGGERO Margherita Universita di Torino, Italy, [email protected]

• ROMERTim Universitat Essen, Germany, [email protected] • ROSSI Maria Evelina Universita di Genova, Italy, [email protected] • SALMON Paolo Universita di Bologna, Italy, [email protected]

• SANGARE Daouda [email protected] • SANGHARE Mamadou Dept. Maths et Info. FST, Universit de Dakar, Senegal, [email protected] • SBARRA Enrico Universitat Essen, Germany, [email protected]

• SKOLDBERG Emil Stockholm University, Sweden, [email protected] • SMITH Karen University of Michigan, USA, [email protected] . SNELLMAN Jan Laboratoire GAGE, Ecole Polyt. Palaiseau Cedex, France, [email protected]

• SRINIVASAN Hema University of Missouri, Columbia- USA, [email protected] . STRANO Rosario Universita di Catania, Italy, [email protected]

Participants

• TARTARONE Francesca Universita di Trieste, Italy, [email protected] • TBRAI Naoki Saga University, Japan, [email protected]

• TYC Andrzej University of Torun, Poland, [email protected]

• UTANO Rosanna Universita di Messina, Italy, [email protected]

. VALABREGA Paolo Politecnico di Torino, Italy, [email protected] • VALENZANO Mario Politecnico di Torino, Italy, [email protected]

• VALLA Giuseppe Universita di Geneva, Italy, [email protected]

• VERRA Alessandro III Universita di Roma, Italy, [email protected]

• VETTER Udo Universitat Oldenburg, Germany, [email protected]

• WEIBEL Chuck Rutgers University, USA, [email protected]

• YANAGAWA Kohji Osaka University, Japan, [email protected] • ZAPPALA Giuseppe Universita di Catania, Italy

• ZARZUELA Santiago Universitat de Barcelona, Spain, [email protected]

. ZATINI Elsa Universita di Geneva, Italy, [email protected]

xix

On free complete intersections MARGHERITA BARILE Dipartimento di Matematica, Universita degli Studi di Bari, Via Orabona 4, 70125 Bari (ITALY) MARCEL MORALES Universite de Grenoble I, Institut Fourier, UMR 5582, B.P.74, 38402 Saint-Martin D'Heres Cedex, and IUFM de Lyon, 5 rue Anselme, 69317 Lyon Gedex (FRANCE)

APOSTOLOS THOMA Department of Mathematics, University of loannina, loannina 45110 (GREECE)

1

INTRODUCTION

The determination of the minimum number of equations needed to define an algebraic variety V set-theoretically or ideal-theoretically is an old and important problem in Algebraic Geometry. The defining ideal of a toric variety is generated by binomials. In [1] we have proven that in characteristic zero a simplicial toric variety is a set-theoretic complete intersection on binomials if and only if it is a complete intersection on the same set of binomials. In this paper we want to give a numerical criterion which characterizes a special class of complete intersection toric varieties in characteristic zero. Let K be a field of characteristic 0. Let ei,...,e n be the elements of the canonical basis of -Zf1, and for all i = 1, . . . ,r let aj = (0*1,1, • • • ,ai,n) € JV". Let di ...,dn e Wand set T = {diei,... ,d n e n ,ai,...,a r } C j8V™. Define ..,xn,yi,...,yr] ->• K[ti, . . . ,tn] as the homomorphism of .FC-algebras for which

(xi) = td;ei

<j>(yi) = tai

for alH = 1, . . . , n,

for alH = 1, . . . , r,

where ta' = t"''1 ...tn''". Then Ker = IT is the simplicial toric ideal of T and its affine variety V = V(/r) of zeros in A^+r is an affine simplicial toric variety in the sense of [2, 7], which also includes non normal varieties. The image (K[xi ,...,xn,yi,..., yr}) is the affine semigroup ring K[T] of T. In other terms,

Barile, Morales and Thoma

IT is the defining ideal of the variety V in A^+r having the following parametrization: Xl

= ui

1* dsji

—— ——

It U-n

=

UI M ...u^ 1 -"

yi

i/T"

5_

* * "

fi

Let ^VT = {/idiei + • • • + / n + r a r : l\,..., ln+r € JN} be the affine semigroup generated by T and 2ZT = {hdiei + • • • + ln+ra.r '• h, • • •,ln+r € ^} the lattice spanned by T. It is well-known that M(!T) ~ r; it is a complete intersection if and only if it is generated by r elements. We recall the following:

DEFINITION 1.1 Let TI and T2 be non-empty subsets ofT such that T = TI U T2 and TI nT2 = 0. Then T is called a gluing ojT\ and T% if there is a nonzero element a e JNTi n IVT2 such that 2Za = 22Ti n ^T2. The concept of semigroup gluing was defined by J.C. Resales in [6] and used by K. Fischer, W. Morris and J. Shapiro in [3] to characterize all complete intersections affine semigroups. They proved the following: THEOREM 1.2 Let 1NT be an affine semigroup which is not a free abelian semigroup. Then 1NT is a complete intersection if and only if there are two subsets TI and T-2 of T such that T is the gluing of TI and T2 and JNTi , ^VT2 are complete intersection sub semigroups. In this paper we want to treat the case of free complete intersections, a notion due to P.A. Garcia-Sanchez and J.C. Resales (see [4]). They give the following characterization:

THEOREM 1.3 The affine simplicial semigroup JNT is a free complete intersection (with respect to the sequence aj., . . . , ar) if and only if for all i = 1, . . . , r — 1, , . . . ,d n e n ,ai,...,ai}

is the gluing of T^i = {diei,...,dnen,ai,...,a.i--i} and {a-,}.

REMARK 1.4 (a) Suppose that T is free with respect to the sequence ai, . . . , a r , Since JV^ej., . . . , d n e n ) is a complete intersection, it follows recursively from Theorem 1.2 that the semigroup

is a free complete intersection, in particular V is a complete intersection (we shall say that V is a free complete intersection). More precisely, if m^a; € -#VT,_i (wij ^ 0),

On free complete intersections

3

is such that W(mjaO = J^fTi-i n Wa;, and m^ai = £)"=1 HijdjBj + Y^i "i,*^, then setting, for all i = 1 , . . . , r, /?. — ,/ m < _ a-i T-^-1 ... 3-r »./*;,.• .."a .. ..."'.'-i S/l i/i-l '

- f z — J/i

we have that 7T = (F 1 ...,F P ). (b) It follows that for all i = 1 . . . , r one has that

ITr\K[xi,...,xr,yi,...,yi] - (Fi,. ..,#). 2

THE MAIN RESULT

The definition of free complete intersection strictly depends on the ordering. This is shown in the next EXAMPLE 2.1 Let T = {(4,0), (0,4), (2,2), (3,3)}. Then 1NT is free with respect to the sequence (2,2), (3,3). In fact

2(3,3) =3(2,2) generates ^((4,0), (0,4), (2,2)) n ^(3,3), so that T is the gluing of {(4,0), (0,4), (2,2)} and {(3,3)}. The toric variety V associated to T is parametrized

by xi = u\,

x2 = u\,

2/1 = u\u\,

2/2 = u\ul,

it is a complete intersection on the binomials

yi-xix-z,

2/2-2/1-

T is not, however, the gluing of {(4,0), (0,4), (3,3)} and {(2,2)}. In fact

(2,2) = -(4,0) -(0,4) + 2(3,3), but

(2,2)gW((4,0),(0,4),(3,3)). Hence INT is not free with respect to the sequence (3,3), (2,2). LEMMA 2.2 Suppose T is free (with respect to the sequence ai,... ,a r j. The notations are those introduced in Remark 1.4. For alii = 1,... ,r let Mi be a monomial not divisible by yt but divisible by Xj for at least one j, such that Gi = y™' —Mi € IT, where m; ^ 0 and I^m^O = IVT;_i n Was. Then IT = (Gl,... ,G r ).

Proof. Set J = (Gi . . . , Gs). By induction on i = 1,..., r we prove that Fj € J. For all i = 1 , . . . , r set ri 1 Ti r M-l — rai'1 • • • 7x-ncri'"'7/ ivi — Xj yl ' •• •• •ii -yr ' .

First suppose that M\ only involves some of the variables x\... ,xn. Then

Barile, Morales and Thoma

3=1

Since dielt . . . ,dnen are linearly independent, it follows that o~ij = /j,ij for all j = 1, . . . ,n, so that Fj = GI. Now suppose that MI involves some variable y^. Let s be the greatest index i with this property. Then

It follows that r ljS = \sms for some As € W. Now

hence

"i •— 2/1 /~"

.__

—

—

,.mi

(7, -

i

• • • ar,n

0 d

••• 0

0 0 ••• d J

Set c= We shall use the following notation: if A is a matrix with integer coefficients, by

| .A | we shall denote the greatest common divisor of its maximal minors. If all the maximal minors are equal to 0, we set \A\ — 0. By a well-known lemma of number theory (see [5]) the system (**) has an integer solution if and only if |A| ^ 0 and \A\ — \Ac . Now:

\A\= gcd (M* d, we prove that the number of ideals of colength h is a polynomial of degree d — 1 in h, if h > IR(&/R : R) (cf. Theorem 3.8). In particular, when d = 1, i.e. when

R is analytically irreducible, the number of ideals of colength h is, for large h, a constant that of course depends on the cardinality of the residue field of R.

All the information on the number of ideals of finite colength of a ring R can be collected in a generating function, the colength series of R, which in the case of our subclass of rings has a nice form. To prove the mentioned results, we use the value semigroup associated to a one-dimensional analytically unramified ring and we refer for that to [1].

11

12

2

Barucci and Froberg

GENERALITIES

Let R be a (not necessarily Noetherian) ring. We consider ideals J of R of finite colength (i.e. ln(R/J) — h < oo). This is equivalent to say that R/J is an Artinian ring.

Notice that many ideals of finite colength may exist also in non-Noetherian rings. R = Z + XQ[[X]] is an example of a non-Noetherian ring with ideals of any colength h € N. As a matter of fact .R/XQ[[Jf ]] = Z and so, if p is a prime

in Z, phR = ph1 + X^[X]} is an ideal of R of colength h. LEMMA 2.1 Suppose that J C I are ideals of a quasilocal ring (R,m) with 1R(I/J) = 1. ThenlmC J. Proof. If / is finitely generated, this follows from Nakayama's lemma, but the statement is always true. Let t € / \ J. Then t £ J + tm, since otherwise t — j + tmi,j € J, mi € m, so i(l — mi) = j € J, and t £ J since 1 — m\ is invertible. Since J C J + tm C I and the last inclusion is proper, the first inclusion cannot be proper, and we get J 4- tm = J for all t € /, so Im C J.

PROPOSITION 2.2 Let (R, m) be a quasilocal ring. If J is an ideal of colength h, then mh C J. In particular J is m-primary. Proof. We use induction on h. If h — 1, then J — m. Suppose ln(R/J) = h and that the statement is proved for ideals of colength h — 1. Choose an ideal / 3 J of colength h - 1. Then mh~l C /, so mh C ml C J by Lemma 2.1.

COROLLARY 2.3 Let R be a ring with a finite number of ideals for each colength h € N and let m be a maximal ideal, then the localization Rm has also a finite number of ideals for each colength h € N.

Proof. By Proposition 2.2, ideals in Rm of finite colength are m.Rm-primary, and there is a 1-1 correspondence between m.Rra-primary ideals Qm in Rm and m-primary ideals Q in .R, and lRm(Rm/Qm) = If the ring (R,m) is Noetherian, then each m-primary ideal is of finite colength, but in general this is not true. By Proposition 2.2, if the maximal ideal of R is idempotent, i.e. m — m2 (this happens for example in a one-dimensional non-discrete valuation domain), then the only ideal of finite colength is the maximal ideal, but each non-zero ideal is m-primary. However, if we restrict to Noetherian rings, we get: PROPOSITION 2.4 Let (R,m) be a local (i.e. quasilocal and Noetherian) ring

of dimension > 0. Then there exists, for each h € N, an ideal of colength h. Proof. By induction on h. Let I be an ideal of colength h — 1. Any ideal J which is maximal in the set of proper subideals of J is of colength h.

In the sequel we will restrict to (Noetherian) local rings. There is no restriction to assume that R is complete:

The number of ideals of finite colength

13

PROPOSITION 2.5 If (R,m) is local with (m-adic) completion (R,m), there are just as many ideals of colength h in R as in R. Proof. By Proposition 2.2, mh C / if / is of colength h (and correspondingly for ideals in R), and R/mh ** R/mh.

Notice however that even such a simple ring as R — C[X, Y]/(X, Y)2 = C[x, y] has infinitely many ideals of colength 2. Any maximal chain of ideals in R looks like this:

R3(x,y)D(ax + by)l>(Q)

and there are infinitely many choices for (a, 6) ^ (0,0) giving different ideals. The following proposition gives the class of rings we will study. PROPOSITION 2.6 Let (R,m) be a local ring. Then, for each h 6 N, there is a finite number of ideals of colength h if and only if R is a DVR, an Artinian principal ideal ring, or if R/m is finite.

Proof. Suppose that the number of ideals of colength 2 is finite. The ideals of colength 2 are in 1-1 correspondence to .R/m-subspaces of m/m2 of codimension 1. Then either m/m2 is one-dimensional or R/m is a finite field. In the first case m = (x) is a principal ideal and, since by Krull intersection theorem fli>o m* = 0, we get that every element of R is of the form ex1, for some i > 0 and some unit e. It follows that, if ml ^ 0 for each i, then R is a DVR and, if m1 = 0 for some i, R is an Artinian principal ideal ring.

If R is a DVR or an Artinian principal ideal ring, the number of ideals of each colength is at most one, so we assume that R/m is a finite field. By induction we can assume that there are finitely many ideals Jj of colength h — 1. The ideals of colength h corresponds to .R/m-subspaces of Ji/mJi of codimension 1, which are finitely many. We are interested in the growth of the number of ideals of colength h in

a local ring R as a function of h. We denote the number of ideals in R of colength h by OH(/I), or just ft(/i) if the ring R is understood from the context. If R is Artinian, then fi(/i) = 0, if h » 0. We will first see that, if dim/? > 2, then fl(h) cannot be bounded by a function which grows less than exponentially,

thus the following theorem shows that it is natural to restrict to one-dimensional rings. We will use the following, certainly well known, lemma.

LEMMA 2.7 Let V be a vector space of dimension n over a field with q elements. Then the number of subspaces of dimension (or codimension) [n/2] is at least

Proof. The statement is trivially true for n = 1. If n > 1, the number of subspaces of dimension [n/2] is (qn - l)(qn - q) • • • (qn - qln/2l-l)/((q^n/^ l)( g [n/2] _ q ) . . . ( q [ n / 2 ] _ g [n/2]-i)) n

2

1

n 2

=

flgg)-1 (q» - g')/(gl n / 2 I -

subspaces of codimension Mn in mn/mn+l, so there are at least g(M") ideals of colength an2 + bn + c + Mn in R. If we let M'n = an2 + bn + c + Mn, we will show that (M n ) 2 > FM'n for some F > 0, if n » 0. Since (Mn)2 > (an)2 /I if n » 0 and M'n < 2an2 if n » 0, it suffices to show that (on) 2 /2 - F • 2an2, for some F > 0, and so F = a/4 will do. If dim R > 2, let / be an ideal in R such that dimR/ 1 = 2. Obviously fi^(n) > £lR/j(n). In the next section we will see that we can get good control over the growth of fl(/i) for a large class of one-dimensional rings.

3

ANALYTICALLY UNRAMIPIED ONE-DIMENSIONAL RINGS

In this section we consider a particular class of one-dimensional rings. In all this section R will be an analytically unramified one-dimensional local ring, i.e. a one-dimensional reduced Noetherian local ring, such that the integral closure R is finite over R. An important class of examples of such rings are the local rings of an algebraic curve. As we noticed in the previous section, it is not restrictive to suppose that R is complete. So we can suppose that, if PI, . . . ,Pd are the minimal primes of R, each R/Pi is analyticlly irreducible, with integral closure Vi, a DVR. Thus we have R C R/Pl x • • • x R/Pd and R = Vi x • • • x Vd. We also suppose that R is residually rational (i.e. that all localizations at maximal ideals of R have the same residue field as R) and that the cardinality of the residue field of R is at least equal to the number d of minimal primes. Since R C R/P\ x • • • x R/Pd C Vi x • • • x Vd, each element x = (x\ , . . . , Xd) 6 R has a value v(x) = (vi(xi),... ,vd(xd)), where, for i = 1,... ,d, v^ is the valuation of the DVR Vi (it is convenient to assume Uj(0) = oo). The value semigroup of R is 5 = v(R) = {v(x); x E R} C (N U {oo})d and each ideal I C R has its value set v(I] = {v(x);x G /} C S. On 5 there is a natural partial ordering, (QI, . . . ,a<j) < (/Si, . . . ,/?d), if Q-i < /?« for all i. For other properties of S, we refer to [1]. If C = (R : R) is the conductor, then C is an ideal (of R and) of R , so C = ti5lVi x • • • x td6dVd, where tt is the uniformizing parameter of Vj. Thus v(C) = {a = (ai,... ,ad) e (NU{oo}) d | a» > Si, for i = 1, . . . ,d}. We will' always denote minv(C) by 6 = (Si,.. . ,Sd) in the sequel. Notice in particular that each element x £ R = V\ x • • • x Vd, with v(x) > S (i.e. v(x) 6 v(C)) is in R, because it is in C. If J C / are ideals of R, it is possible to compute IR(! / J) looking at v(I) and v(J) (cf. [1, Section 2.1]).

The number of ideals of finite colength

15

Finally, in order to study how the number of ideals of colength h grows with h in R, we have to suppose that the residue field of R is finite, cf. Proposition 2.6. In this setting, that is fixed for all Section 3, and with the notation introduced above, if 7, J are ideals of R, we define I ~ J if there exists an element x in the quotient ring of R such that v(I) = v(xJ). This is an equivalence relation and we call a shape for the ideals of R an equivalence class. If / is an ideal in the equivalence class I, we say that I is the shape of I or I is of shape I. Notice that the shapes are finitely many for a ring R.

EXAMPLE 1. For the ring R = k[[(t, u), (t3,^)}} = k[[x, y}}/(x3 - y) n (x2 - y) that has the following value semigroup

1 2 3 Fig. 1. The value semigroup of R we have the following shapes:

Zi =the principal shape. Z 2 =shape(((£,u), (£ 3 ,u 2 )).R). 23=shape(C). DEFINITION. Given a shape 1 for the ideals of R, we define the function ftz(/i) as the number of ideals of R of shape I and colength h. Of course we haven(/i) = Ei fi i( ft )-

3.1 The analytically irreducible case We first consider the analytically irreducible case, i.e., we assume that the integral closure of the ring R is a DVR which we denote by (V, i). We denote the conductor C = R : V by t5V.

16

Barucci and Froberg

LEMMA 3.1 The map &(/) = t*I,i > 0, from ideals I with minv(I) = S to ideals J with minu(J) — i + 6 is a bisection which preserves the shape of the ideal. Proof. Since tll is a fractional ideal and fl C C C R, we get that (j>i(I) is an ideal of R. The map is bijective with (j>^l(J) = t~*J as inverse. The shape is preserved by the definition of shape.

LEMMA 3.2 fi z (/i) is constant, ifh>6. Proof. If minw(/) < S, then 1R(R/I) = #(w(H) \ «(/)) = #((v(R) \ «(/)) n

[1, 6)} + #((v(R) \ v(I)) n [S, oo)) < 1R(R/C) + 1R(V/K) = 1R(V/C) = 6. Thus 7 C C, if 1R(R/ 1) > S. According to Lemma 3.1, the number of ideals of shape X is constant (i.e. independent of minu(/)) for all ideals inside the conductor.

We now state the main result for analytically irreducible rings. PROPOSITION 3.3 // R is analytically irreducible, then fi(/i) is constant, if

h > 1R(V/C). Proof. We have ft(/i) = J^x^zW- The sum is finite and each summand is a constant, by Lemma 3.2, if h > 6 = 1R(V/C). As usual it is convenient to collect the information in a generating function. We define the colength series of R to be CLR(Z) = ^%L0ft(h)Zh. For an analytically irreducible ring R we get CLR(Z) = p(Z)/(l - Z). Then p(Z) 6 Z[Z], and p ( l ) = Q(/i), for h > 1R(V/C). The constant fi(/i),/i » 0, of course depends on q — \R/m\. We will determine this dependence in an example.

EXAMPLE 2. Let R = k [ [ t 3 , t 4 , t 5 ] } . There are the following shapes of ideals: Ii = shape(-R), Z2 = shape((£ 3 ,£ 5 )), I3 = shape((* 3 ,i 4 )), and finally I4 = sh&pe((t3,t4,t5)). We get, if q - \k\,

n(2) = nl2(2) + nls(2) + nl4(2) = q + q2 + 1,

n(ft) = SlXl(h) + O l2 (/i) + ft l3 (/i) + fi l4 (/i) = q2 + q + q2 + 1, if h > 3. CLR(Z) = (l + (q + q2)Z* + q2Z3)/(l - Z).

We could generalize the example and show that, if v(R) = (6,6+1,... ,26 — 1), then the constant tl(h) is a polynomial of degree [£/2]2 in g = \R/m\, if h > 6. In general the dependence of fl(/i) of q is more complicated. We can,

however, show that ft(/i) is always bounded by a polynomial of degree [S/2]2 in

q3.2 The non-analytically irreducible case We consider now the analytically unramified case with d > 1. Recall that, as above, S = (Si,... ,(7) e Tslt...,s,,0.+i,...,i3d does not depend on the element a — minu(7).

Denote the constant in Lemma 3.6b) by fi(Tslt...,s,,f3,^l,...,i3d)For an analytically unramified ring with d > 1 minimal primes, we need to make the definition of the function flj(/i) finer.

DEFINITION. Given a shape I for the ideals of ft, we define the function uz(h,Tglt,..ts,,p.+i,...,0d) as the number of elements a £ Tglt_,^s,,/3,+i,...,0d such that there exists an ideal 7 in ft of shape I and colength h, with minv(7) = a.

18

Barucci and Froberg

The computation of the function i^i(h,Ts1,...ts,,^,+1,...,ffd) gives an answer to our problem at semigroup level and is the first step in the computation of the growth (with h) of fi(ft), the number of ideals of colength h. It is convenient to introduce before next lemma another notation. If I is an ideal of shape I, with minv(J) = mmTSl,...,s,,/3,+1,...,0d, set bz(TSl,... ,s.,p,+i,...,pd) = ln(R/I)-

LEMMA 3.7 ui(h,Tglt...tgftpi+li...tpd)

is a polynomial of degree at most s - I ,

Proof. We know that Tgli...tgttftt+l!...>0d is a finite union of subsets T of type (1) described in the beginning of this section. We will first count the number of elements a € T (where T is of type (1)), such that there exists an ideal / of R of shape X and colength ft, with minu(J) = a. If such an ideal / with minf(I) £ 2 1 < 5 1) ... ) j 4) ^ +1> ... $d exists, we have to count the number of ways to write ft — bz(Tglt...tgft/3f+lt...tpd) as a sum of u non-negative summands ft, = di —Si, where u < s. This is given by 'ft- &zCT